In an earlier post I discussed the benefits that shoplifting could have on inflation. In a nutshell, my thesis defended the (theoretical) use of ‘moderate’ shoplifting as a way to fight against inflation. Indeed, if the share of stolen goods grows, retailers have to anticipate this behaviour by increasing their prices in order to offset the losses incurred because of theft. I nonetheless ended up moderating my remark by underlining the fact that ‘too much theft could kill theft’. The topic of this post is actually to prove this latter assertion. To do so, I will rely on a powerful and widespread microeconomics tool called ‘game theory’.
According to Wikipedia, game theory is “the study of mathematical models of conflict and cooperation between intelligent rational decision-makers”. Although historians can find traces of rudimentary game theory dating back to the 18th century, this theory started to be properly formalised appeared in the 1920s, championed by brilliant economists such as von Neumann and later John Nash – made famous in the eyes of the general public through the movie A Beautiful Mind. Although the framework sometimes requires ample simplification to be quantitatively workable, it yields satisfactory results in our case.
Note: The following paragraphs may repel non-scientific readers. Those readers may prefer to jump directly to the conclusion, which is signalled by .
Let us represent today’s problem as a game between two players, namely the shoplifter community and the shop – we use one representative shop and we similarly consider that the shoplifting community can be considered as a relatively homogeneous group. We assume that both players are risk-neutral, i.e. all they care about is their utility at the end of the game, irrespective of the degree of uncertainty surrounding this outcome. The ‘sequential game’ is the following:
- First, the shop decides whether to set-up a surveillance system (CCTV, security guards etc. at total cost C>0) or not (at no cost).
- Having observed this choice, the shoplifter decides whether to try to steal or not. The shoplifter’s probability of success depends on the presence of a surveillance system. If there is none, this probability is pmax. If there is one, this probability falls to pmin, where pmin<pmax.
- Finally, we distribute the rewards. If the shoplifter is successful, he enjoys the product of his theft, assumed to be equivalent to $ in dollars – the same amount is withdrawn from the shop’s utility. If he is caught, he has to face a penalty equivalent to a cost of F – although the penalty may be made of non-monetary items e.g. prison sentences.
We assume F, pmin, pmax and C to be known and constant throughout the problem. The question we try to answer is: to which extent does the outcome of this problem depend on $?
To solve sequential games, game theory uses the principle of ‘backward induction’. We start by deducting the optimal solution for the agent playing last, and we move backward to anticipate each player’s move given the subsequent decisions made by the other players. Here, the shoplifter plays last, so we will pay attention to him first.
The shoplifter will try to steal only if his expected utility is greater than the utility he gets by staying home, which we assume to be equal to 0. Mathematically speaking, this translates into:
- if the shop is equipped or
- if the shop is not.
We can rewrite the previous two equations as follows: and . In proper English, these equations illustrate the fact that the shoplifter will only try to steal if the reward is high enough compared with the potential punishment.
If we plot the shoplifter’s decision as a function of $, we come up with the following strategy:
We now turn our attention to the shop, which can perform the same reasoning as the one we just did and is therefore able to predict the shoplifter’s behaviour depending on all the problem’s variables. We can distinguish three cases:
- If the shoplifter does not try in all cases because the reward is not worth the effort, then it is optimal for the shop not to buy protection.
- If the shoplifter tries only if the shop is not equipped, it is optimal for the shop to protect itself if the expected savings of protecting the shop against theft are greater than the cost of the equipment itself. If the equipment is indeed prohibitively expensive, the shop may be financially better-off letting the theft unmonitored. Mathematically this can be written as: or . Note that the boundary does not depend on F, i.e. the shop makes his decision irrespective of the legal framework.
- If the shoplifter tries in all cases, then there is a possibility that the shop pays for the equipment and gets stolen from. The related equation is therefore or .
Although we could continue to solve the game using abstract variables, the conclusion is more powerful if we now switch to a numerical illustration.
Let us set the fine F to £500, the cost of equipment C to £200, the probability of successful theft without any equipment to pmax=80% and the probability of successful theft with equipment to pmin=5%. We can now replace the formulaic boundaries driving the shoplifter’s behaviour in the axis above by their numerical values, respectively £125 and £9,500. The latter value can appear as very high, but is only due to the fact that the shoplifter has a very high chance of being detected and will therefore only try his luck if the reward significantly outgrows the (almost certain) penalty.
For the shop, the reasoning is as follows:
- “If the expected reward is less than £125, the theft will not even try so I do not spend any money on surveillance”.
- “If the reward is between £125 and £9,500, then the theft will only try if I am unprotected. On my side, I am better-off on average by setting up a surveillance system only if the expected take is greater than .”
- “If the reward is greater than £9,500, then the theft will always try. On my side, I am better-off on average by setting up a surveillance system only if the expected take is greater than . This condition is always verified given that we are only considering takes greater than £9,500 in this third case.
Note: The non-scientific reader may resume from here. That makes the article quite shorter I must admit.
As a summary, if we put the decision of the theft and the decision of the shop together as functions of $, our conclusion is the following:
How can we interpret those results?
- If the expected take is too small (smaller than £125 in our example), the fine is relatively too high for the shoplifter to take the risk. In this first case, the shop is actually protected by the legislation around shoplifting.
- If the expected take is high enough, i.e. in our example between £125 and £250, then the law does not provide a strong enough deterrent while the equipment is relatively too costly for the shop given the expected loss it faces. Within this ‘window of opportunity’ the optimal choice for the shoplifter is actually to try his luck.
- If the expected take is between £250 and £9,500 the shoplifter will not try: his probability of success is too low given the implementation of the surveillance system.
- Finally, if the expected take is greater than £9,500, the shoplifter is willing to take all possible risks – but, realistically, how likely is the shoplifter to manage to steal £9,500 worth of goods with only a 5% chance of being detected?
Game theory here shows us that, provided that the shop can perfectly anticipate the value of $ and both players are rational and risk-neutral, there is indeed a gap where attempted shoplifting makes rational sense for everyone. Nonetheless, as already pointed out in my earlier post, economists have managed to translate rationality in their models, but morality has been so far largely left behind.